Simple Barban--Davenport--Halberstam type asymptotics for general sequences

TL;DR

This paper establishes elementary estimates for the variance of general sequences in arithmetic progressions, leading to asymptotic results for smooth numbers, thus extending understanding of distribution patterns in number theory.

Contribution

It provides new elementary estimates for variance in arithmetic progressions applicable to general sequences, including smooth numbers, with potential for asymptotic formulas.

Findings

Derived variance asymptotics for smooth numbers less than x.

Provided elementary proofs for variance estimates in arithmetic progressions.

Addressed a question by Granville and Vaughan on distribution of smooth numbers.

Abstract

We prove two estimates for the Barban--Davenport--Halberstam type variance of a general complex sequence in arithmetic progressions. The proofs are elementary, and our estimates are capable of yielding an asymptotic for the variance when the sequence is sufficiently nice, and is either somewhat sparse or is sufficiently like the integers in its divisibility by small moduli. As a concrete application, we deduce a Barban--Davenport--Halberstam type variance asymptotic for the $y$-smooth numbers less than $x$, on a wide range of the parameters. This addresses a question considered by Granville and Vaughan.